A group consists of 4 couples in which each of the 4 persons have one wife each. In how many ways could they be arranged in a straight line such that the men and women occupy alternate positions?

Options

A. 1152
B. 1278
C. 1296
D. none of these

Answer

To solve the problem of arranging 4 couples (8 people) in a straight line such that men and women occupy alternate positions, we can break it down into a series of steps.

### Step-by-Step Solution:

1. **Identify the Arrangement Pattern**:
Since we need to arrange the men and women in alternate positions, we have two possible arrangements:
- Case 1: Man (M), Woman (W), Man (M), Woman (W), Man (M), Woman (W), Man (M), Woman (W)
- Case 2: Woman (W), Man (M), Woman (W), Man (M), Woman (W), Man (M), Woman (W), Man (M)

2. **Calculate Arrangements for Case 1 (M-W-M-W-M-W-M-W)**:
- There are 4 men who can be arranged among the 4 positions designated for men. The number of ways to arrange 4 men is given by \(4!\) (4 factorial).
- Similarly, there are 4 women who can be arranged among the 4 positions designated for women. The number of ways to arrange 4 women is also given by \(4!\).

Therefore, the total arrangements for Case 1 is:
\[ 4! \times 4! = 24 \times 24 = 576 \]

3. **Calculate Arrangements for Case 2 (W-M-W-M-W-M-W-M)**:
- The calculation is the same as in Case 1. The number of ways to arrange 4 women in the women’s positions is \(4!\), and the number of ways to arrange 4 men in the men’s positions is also \(4!\).

Therefore, the total arrangements for Case 2 is:
\[ 4! \times 4! = 24 \times 24 = 576 \]

4. **Combine Both Cases**:
Since both cases are valid and mutually exclusive, we can add the total arrangements from both cases:
\[ \text{Total arrangements} = 576 + 576 = 1152 \]

5. **Final Consideration**:
There is no additional arrangement required for the positioning of men and women since they are already defined by the cases. Therefore, the final answer remains 1152.

### Final Answer:
The total number of ways to arrange the 4 couples in a straight line such that men and women occupy alternate positions is **1152**.